Thermal Performance-Driven Simulation and Optimization of Tessellated Façade Shading Systems in Mediterranean Educational Buildings

Abstract

Despite the growing use of tessellated and patterned façades in contemporary architecture, their thermal performance, particularly in cooling-dominated educational buildings, remains insufficiently quantified, with existing studies largely prioritizing daylighting or aesthetic outcomes over energy-driven thermal behavior. This study aims to systematically evaluate how different tessellated façade geometries and perforation ratios influence thermal performance and cooling demand in a Mediterranean climate, and to identify an optimal façade configuration that balances multiple thermal objectives. Three tessellation typologies—nature-inspired (Voronoi), Islamic geometric, and folded origami-based patterns—were parametrically generated and applied as external shading screens to an educational building. Annual thermal simulations were conducted using Climate Studio to assess four performance metrics: solar heat gain, energy use intensity, hours of overheating derived from operative temperature, and peak cooling demand. A post-simulation, data-driven, multi-objective, decision-support approach was applied using Compromise Programming to systematically evaluate and rank discrete façade alternatives based on multiple thermal performance criteria. Results indicate that all tessellated façades reduce solar heat gain and peak cooling demand relative to the unshaded baseline, with performance strongly dependent on both geometry and perforation ratio. Lower perforation ratios (20%) consistently outperform more open configurations, while Voronoi-based façades achieve the most balanced overall thermal performance across all evaluated criteria and emerging as the top-ranked solution. The study’s novelty lies in its comparative, cooling-focused evaluation of fundamentally different tessellation logics using transparent, decision-oriented optimization rather than subjective comfort indices or computationally intensive evolutionary algorithms. Beyond its specific findings, the research provides a transferable methodological framework for integrating geometry-informed façade design into early-stage decision-making, supporting climate-responsive and energy-efficient educational architecture in Mediterranean and similar climates.

1. Introduction

The thermal performance of a building refers to the ability of its envelope, including walls, roof, windows, and façades, to regulate internal temperature by minimizing heat loss during cold periods and reducing heat gain under warm conditions [1]. Optimizing thermal performance is crucial for achieving energy efficiency [2], reducing dependence on mechanical heating and cooling systems [3], and ensuring occupant comfort [4]. Key determinants include insulation properties [5], material conductivity [6], ventilation strategies [7], and solar gain control [8]. As buildings account for approximately 30–40% of global energy consumption, enhancing their thermal performance has become a central priority in sustainable architectural design and environmental policy [9]. Improving envelope performance not only reduces operational energy demand and carbon emissions but also enhances long-term occupant wellbeing and resilience against climate variability [10,11]. Recent studies also highlight that advanced façade strategies can deliver significant energy savings by optimizing window-wall ratios and dynamic envelope responses to solar radiation [12].

Tessellation and geometric patterning have long been integral to architectural design, especially in façades, due to their capacity to merge structural efficiency, material economy, and aesthetic richness [13]. Tessellation involves the repetition of geometric modules, such as triangles, hexagons, or biomimetic patterns, to create continuous and visually coherent surfaces [14]. These configurations influence key façade parameters, including the distribution of openings, surface textures, and shading elements, thereby modulating the building’s interaction with solar radiation, daylight, and airflow [15]. Recent advances in parametric modeling, digital fabrication, and performance simulation tools have enabled designers to generate and test complex tessellated geometries that dynamically respond to climatic conditions [16]. As a result, tessellated façades have evolved from purely decorative systems into adaptive, performance-driven architectural components [17].

Within contemporary research, the intersection between tessellated façade geometry and building envelope thermal efficiency has emerged as a promising area of inquiry. Numerous studies have explored the environmental potential of tessellated façades, particularly in optimizing daylighting [18], visual comfort [19], and natural ventilation [20]. In addition to these general trends, several studies have investigated the role of façade design and tessellated shading systems in regulating building envelope performance, particularly in terms of solar radiation control and energy efficiency. For instance, Hachem and Elsayed [21] demonstrated that façade system configurations significantly influence building energy performance, while Bande et al. [22] reported notable reductions in energy consumption through parametric façade strategies. Similarly, scholars such as Ningsih et al. [23] highlighted the capacity of geometric façade systems to reduce solar radiation levels through controlled opening configurations.

In terms of indoor environmental quality (IEQ), previous research has shown that façade openings and geometric patterns can enhance visual and thermal comfort conditions, improving occupant well-being [24,25]. From a methodological perspective, existing research predominantly relies on simulation-based approaches and parametric design workflows, as demonstrated in studies such as [19], with limited integration of multi-objective evaluation frameworks or comparative analyses across multiple performance indicators.

However, as highlighted in the systematic review [13], the influence of tessellated façade geometries on thermal performance remains comparatively underexplored, with only 18% of studies addressing this aspect, compared to 73% focusing primarily on daylight performance. In addition, studies explicitly linking tessellated façade design to thermal comfort conditions remain limited. The review further identifies several critical gaps, including a predominant focus on hot-arid climates [26,27,28] (62% of studies), an overrepresentation of office buildings (67%) [29,30], and limited attention to residential, educational, and commercial typologies. Moreover, geometric diversity remains constrained, with hexagonal tessellations receiving disproportionate attention [31,32,33] while biomimetic, Islamic geometric, folding, and origami-based configurations remain under-investigated. These findings highlight the need to broaden both the climatic scope and geometric repertoire of tessellation studies.

Despite these identified gaps, a key methodological limitation remains insufficiently addressed. Existing studies typically evaluate tessellated façade configurations in isolation or focus on single performance aspects, limiting the ability to systematically compare fundamentally different geometric logics under consistent boundary conditions. In addition, there is a lack of structured approaches to assess trade-offs among multiple thermal performance criteria—such as solar heat gain, energy use, thermal comfort, and peak cooling demand—which are inherently interdependent and often conflicting. Consequently, a central scientific challenge is not only to expand the range of geometries and contexts studied, but to establish a systematic and reproducible framework that enables the comparative evaluation of tessellated façade configurations and supports the identification of balanced design solutions under cooling-dominated climatic conditions.

To address these research gaps, the present study investigates the impact of three distinct tessellated façade patterns—nature-inspired, Islamic geometric, and origami-based—on the thermal performance of an educational building located in Cagliari, Italy, representative of the Mediterranean climate. Using a parametric workflow, façades are generated in Rhino–Grasshopper and simulated through Climate Studio for thermal and energy performance analysis. Given the high solar radiation and prolonged cooling seasons characteristic of Mediterranean regions, the study particularly examines the effectiveness of tessellated shading systems for East-facing façades in mitigating solar heat gain and enhancing indoor mitigates overheating risk.

The identified gaps indicate a lack of comprehensive approaches capable of evaluating the combined effects of façade geometry and perforation on thermal performance. Addressing this limitation, the present research seeks to answer three key questions: (1) To what extent can tessellated façade configurations influence building thermal performance under Mediterranean climatic conditions? (2) Which tessellation typology—nature-inspired, Islamic geometric, or origami-based—achieves the highest efficiency in reducing energy use intensity, solar heat gains and cooling energy demand? (3) What is the optimal perforation ratio and geometric configuration for minimizing overheating risk of educational buildings in Mediterranean climate? It assumes that tessellated façades can enhance thermal performance and that optimizing design parameters can further improve efficiency. By advancing climate-responsive façade design strategies that reduce cooling energy demand and mitigate overheating in educational buildings, this study directly contributes to the United Nations Sustainable Development Goals (SDG), particularly SDG 7 (Affordable and Clean Energy) through improved energy efficiency, SDG 11 (Sustainable Cities and Communities) by promoting resilient and climate-adaptive building envelopes, and SDG 13 (Climate Action) by supporting the reduction in operational carbon emissions in the built environment.

This study aims to systematically investigate the influence of tessellated façade geometries and varying perforation ratios on thermal performance and cooling-related energy demand under Mediterranean climatic conditions, with the objective of identifying façade configurations that achieve a balanced response across multiple thermal criteria. The outcomes aim to advance performance-driven architectural design by demonstrating the potential of tessellated shading façades as effective climate-responsive solutions for improving thermal efficiency in Mediterranean regions and comparable climates. This study does not aim to exhaustively explore all façade parameters, but rather to isolate and systematically evaluate two primary geometric drivers, tessellation type and perforation ratio, while maintaining controlled boundary conditions. This approach enables a clear attribution of thermal performance variations to geometric configuration, providing a structured and transferable framework for façade design evaluation.

The remainder of this paper is structured as follows. Section 2 presents the materials and methods, detailing the research framework, simulation tools, performance metrics, and the multi-objective decision-support approach adopted in the study. Section 3 reports the results, including the thermal simulation outputs and a comparative analysis of the thermal and energy performance of the investigated façade geometries across different perforation ratios. Section 4 discusses the findings, interpreting the observed performance trends in relation to façade geometry and climatic context, highlighting the study’s scientific contributions, practical design implications, and associated limitations, and outlining directions for future research. Finally, Section 5 concludes the paper by summarizing the key findings and emphasizing their relevance for sustainable and climate-responsive façade design.

2. Materials and Methods

The methodology of this study integrates a parametric design workflow, simulation-based thermal analysis, quantitative performance evaluation to investigate the impact of tessellated façade geometries on building thermal performance and multi-objective optimization. By coupling computational modeling with environmental simulation, the research establishes a systematic framework for evaluating how different tessellation typologies influence thermal comfort and energy efficiency in a Mediterranean climate. The simulation-based approach adopted in this study follows a deterministic and reproducible framework, where all façade configurations are evaluated under identical boundary conditions using a validated energy modeling engine (EnergyPlus, version 23.1, U.S. Department of Energy, Washington, DC, USA). This ensures consistency and comparability across all scenarios. The selection of variables was intentionally constrained to ensure methodological clarity and to enable a controlled comparative analysis of geometric influence on thermal performance. Expanding the number of variables would increase model complexity and reduce the interpretability of results. The study follows five sequential phases:

(1) Initial Modeling, involving the development of the base-case building model and the identification of façade orientation and climatic parameters; (2) Parametric Modeling, in which three façade configurations, nature-inspired, Islamic geometric, and origami-based tessellations, are generated using Rhino and Grasshopper; (3) Thermal Simulation, performed through Climate Studio to assess thermal behavior under real climatic conditions using thermal metrics; (4) Thermal Performance Evaluation; and (5) Optimization Procedure, aimed at identifying the most efficient façade configuration and perforation ratio that minimize cooling loads while maintaining occupant comfort. The methodological workflow and corresponding procedures for each phase are illustrated and described below (Figure 1).

2.1. Study Area

The case study is situated in Sardinia, Italy (Figure 2), the second-largest island in the Mediterranean basin, which exhibits a warm temperate climate characterized by hot, dry summers and mild, humid winters [34]. This Mediterranean setting is highly relevant for investigations into daylighting and thermal performance, as the region is exposed to high levels of solar radiation throughout the year [35]. Such conditions create a critical design challenge in balancing sufficient natural daylight provision with the control of glare and summer overheating. Moreover, Sardinia’s distinctive geographical context, strong cultural heritage, and growing emphasis on sustainable architectural practices make it a suitable laboratory for testing innovative façade strategies aimed at improving environmental performance [36]. Within Sardinia, the city of Cagliari (Figure 2c), the regional capital located on the island’s southern coastline, was selected as the focus of this study. As a medium-sized Mediterranean city, Cagliari experiences intense solar exposure, which contributes to recurring issues of glare, overheating, and increased cooling energy demand [37].

The selected case study represents a typical educational classroom configuration in Mediterranean contexts, characterized by moderate depth, high occupancy density, and significant dependence on daylight and natural ventilation. Such spaces are particularly sensitive to solar exposure and overheating, making them suitable for evaluating façade shading performance. The east-facing orientation was intentionally selected due to its exposure to low-angle morning solar radiation, which is difficult to control through conventional shading strategies and often contributes to early-day overheating in educational buildings. This condition provides a critical test scenario for assessing the effectiveness of tessellated shading systems under challenging solar exposure conditions.

2.2. Thermal Metrics and Their Role in Educational Buildings

The evaluation of thermal performance in this study is based on a set of quantitative indicators that collectively capture the interaction between façade geometry, indoor thermal conditions, and energy demand. In educational buildings, indoor thermal environments directly influence students’ cognitive performance, concentration, and overall wellbeing, making thermal performance a critical criterion in environmental design [38]. Given that classrooms typically operate during daytime hours and accommodate high occupant densities, façade design plays a central role in regulating solar exposure, moderating indoor temperatures, and reducing reliance on mechanical cooling systems [39].

To assess these aspects, the study adopts four complementary thermal and energy performance metrics. Energy Use Intensity, expressed in kWh/m²/year, is used to quantify annual cooling-related energy consumption and to evaluate the overall energy efficiency implications of different façade configurations [40]. Annual Solar Heat Gain through the façade and glazing is analyzed to characterize solar-driven thermal loads entering the indoor environment, which are a primary contributor to overheating and cooling demand in Mediterranean climates [41].

Indoor thermal conditions are further examined through Hourly Operative Temperature, which captures the combined effects of air temperature and radiant heat exchange, enabling a detailed assessment of temporal thermal behavior and the number of occupied hours outside the comfort range [42]. Finally, Peak Cooling Demand, extracted from load duration analysis, represents the maximum instantaneous cooling load required during extreme summer conditions and provides insight into system sizing requirements and peak stress on Heating, Ventilation, and Air Conditioning (HVAC) infrastructure, which is particularly relevant for educational buildings with concentrated occupancy schedules. Together, these metrics, which are presented in

Table 1. Overview of thermal performance metrics used in the study, including their definitions and applications within educational settings.

Metric	Definition	Application in Educational Buildings
Energy Use Intensity	Expressed in kWh/m2/year, Energy Use Intensity measures the total annual energy consumption required for space cooling, reflecting the building’s overall energy efficiency.	Indicates the cooling energy demand in classrooms, assisting in identifying façade configurations that reduce operational energy costs while maintaining comfort.
Solar Heat Gain	Represents the amount of solar radiation transmitted through glazing and indirectly transferred through the building envelope, contributing to the internal heat load of a space.	Evaluates how different tessellated façade patterns mitigate direct solar gain and overheating in classrooms exposed to intense radiation.
Hourly Operative Temperature	Refers to the time-based variation in indoor operative temperature, integrating the effects of air temperature and mean radiant temperature.	Assesses temporal stability of indoor thermal conditions throughout school hours, ensuring consistent comfort during teaching and learning activities.
Peak Cooling Demand	Reflects the building’s sensitivity to solar heat gains, envelope performance, and internal loads.	Represents the maximum cooling capacity required to maintain thermal comfort in educational buildings during peak summer conditions.

, provide a comprehensive and performance-oriented framework for evaluating façade effectiveness without relying on subjective comfort indices. Their integration allows for a robust assessment of how tessellated façade geometries influence annual energy consumption, solar heat mitigation, thermal stability, and extreme cooling requirements.

2.3. Modeling and Thermal Performance Analysis Tools

This study employs Grasshopper (Version 1.0, Robert McNeel & Associates, Seattle, WA, USA), a visual programming plugin integrated with Rhinoceros (Version 8, Robert McNeel & Associates, Seattle, WA, USA), as the core computational environment due to its intuitive interface for generating parametric geometries and its widespread use in architectural performance-based design [43,44]. For thermal performance simulation, Climate Studio (Version 2.0, Solemma LLC, Cambridge, MA, USA), a Rhino-based plugin, was utilized for its seamless integration with Grasshopper and its robust analytical capabilities. Climate Studio provides a comprehensive platform for environmental performance assessment, incorporating energy modeling, thermal comfort evaluation, and solar radiation analysis within a single workflow. The software employs validated energy modeling engines, such as EnergyPlus, to calculate hourly variations in operative temperature, energy use intensity, and solar heat gain, enabling precise assessment of façade-related thermal dynamics [45]. Additionally, Climate Studio allows direct use of local EnergyPlus Weather (EPW) climate data, supporting location-specific thermal simulations from early design exploration to detailed optimization.

2.3.1. Simulation Assumptions and Boundary Conditions

To ensure consistency and comparability across all façade configurations, the thermal simulations were conducted under a fixed set of boundary conditions using Climate Studio (EnergyPlus engine). Internal heat gains from occupants, lighting, and equipment were defined based on standard values for educational buildings, reflecting typical classroom usage patterns. Occupancy schedules followed a daytime academic timetable, corresponding to active use during morning and afternoon hours.

Thermal comfort conditions were evaluated based on operative temperature thresholds defined within the simulation environment. In this study, discomfort hours are identified when the operative temperature exceeds 26 °C, which represents a typical upper comfort limit for mechanically conditioned educational spaces under Mediterranean climatic conditions. This threshold is consistent with commonly adopted criteria in building performance studies and aligns with standards such as ASHRAE 55 [46] and ISO 7730 [47] for cooling-dominated environments. The use of a fixed temperature threshold ensures a consistent basis for comparison across all façade configurations and avoids uncertainties associated with adaptive comfort assumptions. While this approach does not explicitly model occupant-dependent variables such as metabolic rate or clothing insulation, the selected threshold corresponds to typical classroom conditions and is therefore considered representative for comparative analysis.

Operative temperature was calculated directly within the simulation environment (Climate Studio using the EnergyPlus engine), which accounts for both air temperature and mean radiant temperature in accordance with standard heat balance methods. Operative temperature is defined as a combined measure of air temperature and mean radiant temperature, representing the uniform temperature of an environment that would result in the same heat exchange between the human body and its surroundings. In line with established definitions in ISO 7726 [48], it is expressed as a weighted average of air temperature and mean radiant temperature, where the weighting depends on convective and radiative heat transfer coefficients. In the present study, air velocity was assumed to be low and uniform, consistent with typical indoor conditions in mechanically conditioned educational spaces, and therefore its influence on the weighting factors was considered negligible. Consequently, operative temperature values were obtained directly from the simulation outputs without additional post-processing.

Ventilation conditions were defined based on standard assumptions embedded within the simulation environment, corresponding to typical occupancy patterns in educational buildings. Natural ventilation was not explicitly modeled, and the analysis assumes a mechanically conditioned indoor environment. The HVAC system was configured to maintain thermal conditions within a predefined temperature range, with cooling loads calculated using the EnergyPlus simulation engine under ideal load assumptions. The system operates as a simplified, centrally controlled cooling mechanism, activated when indoor temperatures exceed the defined threshold [49], without explicit modeling of detailed system components or control algorithms. This approach allows for a consistent comparison of façade performance by isolating the impact of geometric and material variations, while maintaining uniform boundary conditions across all simulation scenarios.

While indoor environmental quality (IEQ), as defined in standards such as EN 16798-1 [50], encompasses multiple aspects, including air quality and visual comfort, the present study focuses specifically on thermal performance and overheating risk. Accordingly, other indoor environmental quality dimensions, such as indoor air quality and daylight-related visual comfort, are considered outside the scope of this analysis, allowing for a more focused evaluation of façade-induced thermal behavior. All material properties, internal loads, and operational schedules were kept constant across all simulation scenarios, ensuring that observed differences in performance are attributable solely to variations in façade geometry and perforation ratio.

2.3.2. Modeling: Building and the Tessellated Patterns

In the initial phase of the study, an existing educational building located within the Faculty of Architecture and Engineering at the University of Cagliari was selected as the case study (Figure 2). The building’s base geometry was reconstructed in Rhinoceros software to accurately reproduce its architectural form and façade configuration. The digital model included precise representations of volumetric massing, floor-to-ceiling heights, and window-to-wall ratios, ensuring that the dimensions and orientations of openings matched the real structure. To establish valid boundary conditions for thermal performance analysis, a key functional area, a multipurpose classroom, was modeled based on architectural drawings. The spatial layout was carefully reproduced to capture depth, orientation, and surface characteristics, which significantly influence thermal behavior. Additionally, material attributes were assigned to replicate realistic physical properties.

Table 2. Geometric and material attributes of the case study building, including spatial configuration, façade characteristics, and construction materials for the study area multipurpose classroom.

Parameter	Multipurpose Classroom
Function	Teaching and learning space
Floor-to-Ceiling Height	5.00 m
Room Dimensions (L × W)	7.0 m × 4.0 m
Orientation	East-facing
Window-to-Wall Ratio	0.40
Window Dimensions	2.00 m (W) × 2.80 m (H)
Wall Material	White gypsum plaster (U-value: 1.60 W/m2·K)
Ceiling Material	White gypsum plaster (U-value: 1.20 W/m2·K)
Floor Material	Gray mosaic tile (U-value: 2.10 W/m2·K)
Window and Door Frames	UPVC frame with double glazing (U-value: 2.80 W/m2·K)
Shading Device Type	Parametric tessellated screen applied externally

summarizes the geometric and material characteristics of the study area modeled for simulation. These parameters establish consistent boundary conditions and ensure accurate representation of the building’s physical and thermal properties in the analysis.

In the second phase, three distinct tessellation patterns, nature-inspired (Voronoi), Islamic geometric, and folded origami (Figure 3), were developed using Grasshopper. Each pattern was modeled parametrically, allowing for dynamic control of key design variables, particularly the perforation ratio, which governs the degree of solar permeability and shading density. The tessellated screens were applied digitally to the building’s façade, functioning as external shading devices with varying geometric complexity and thermal modulation capacity. This integrated modeling process established the foundation for subsequent thermal performance simulations under real Mediterranean climatic conditions, enabling a comparative assessment of how different tessellation configurations influence heat gain, cooling demand, and indoor comfort. Although tessellation geometry is categorized into three typologies (Voronoi, Islamic, and folding-based), each pattern is generated parametrically and governed by explicit geometric rules that define aperture distribution, edge density, and spatial configuration. Therefore, geometry is treated as a structured design variable rather than a purely qualitative descriptor. The perforation ratios (20%, 40%, and 60%) were selected as representative low, medium, and high openness levels to enable a clear and interpretable comparison of shading performance. This discrete sampling strategy balances computational feasibility with analytical clarity, allowing systematic evaluation of façade permeability without introducing excessive model complexity.

2.3.3. Thermal Performance Simulation

The third phase focused on simulating and evaluating the thermal performance of façade-integrated tessellation patterns using Climate Studio 2.0 within the Grasshopper environment. Each façade configuration was assessed using a consistent set of energy- and temperature-based performance metrics, summarized in Table 1. Simulations were conducted using annual climate data from the EnergyPlus weather file for Cagliari (typical meteorological year). Climate Studio’s energy modeling engine enabled detailed hourly calculations of solar heat gain, zone operative temperature, and building energy consumption.

The simulation workflow follows a structured process linking external climatic conditions, façade configuration, and indoor thermal response. Weather data and boundary conditions are used as inputs to the building model, while the tessellated façade systems modify solar radiation exposure and shading behavior at the envelope level. These variations directly influence heat transfer through the building envelope, which in turn affects indoor environmental conditions and cooling demand. Thermal performance was evaluated using four primary metrics: Energy Use Intensity, Solar Heat Gain, Hourly Operative Temperature, and Peak Cooling Demand. Annual Energy Use Intensity was used as the main indicator of overall energy consumption, allowing direct comparison between façade configurations. Solar heat gain through the façade was quantified to characterize the magnitude of external thermal loads driving cooling demand. Hourly operative temperature distributions and the associated hours outside the comfort range were extracted from zone temperature curves to describe indoor thermal behavior over the year. In addition, Peak Cooling Demand, derived from load duration curves, was used to capture the maximum instantaneous cooling load required during extreme summer conditions, a critical metric for HVAC system sizing in educational buildings. As a result, key performance indicators—including solar heat gain, energy use intensity, hours outside the comfort range, and peak cooling demand—are derived from the simulation outputs, enabling a consistent comparison across all façade configurations.

This phase addressed the first two research questions: (1) how different tessellation patterns influence indoor thermal conditions and cooling energy demand, and (2) which geometric configuration achieves the most effective balance between solar heat reduction and energy performance. By systematically comparing simulation outputs across all façade patterns and perforation ratios, this stage provided a robust quantitative basis for assessing the impact of geometric articulation and façade permeability on thermal performance and energy efficiency in an educational building context.

2.4. Multi-Objective Optimization Framework

Following the completion of the thermal simulations, a comprehensive dataset of energy and thermal performance metrics was generated for all façade configurations, including three tessellation typologies—nature-inspired (Voronoi), Islamic geometric, and folding-based patterns—evaluated at perforation ratios of 20%, 40%, and 60%, together with an unshaded baseline configuration. Each simulated case represents a discrete façade design alternative defined by a specific combination of geometric attributes and associated performance outcomes. Given the discrete nature of the design space and the high computational cost of rerunning detailed thermal simulations, the optimization was conducted as a post-simulation, data-driven decision-support process rather than through iterative simulation-based optimization.

The optimization problem was formulated as a multi-objective minimization task based on four performance objectives, denoted as O1–O4 and summarized in

Table 3. Parameters defining the design space and performance objectives for the data-driven multi-objective optimization of tessellated façade thermal performance.

Category	Parameter	Symbol	Description	Optimization Role
Design Variables	Tessellation pattern type	X	Façade geometric typology, including nature-inspired (Voronoi), Islamic geometric, folding-based patterns, and the unshaded baseline configuration	Discrete decision variable
	Perforation ratio (%)	X	Ratio of open area to total façade area, evaluated at 20%, 40%, and 60% for tessellated façades	Discrete decision variable
Objective Functions	Annual Solar Heat Gain (kWh)	O1	Total annual solar heat gains transmitted through the façade and glazing, extracted from Energy Flow analysis	Minimize
	Annual Energy Use Intensity (kWh/m2·year)	O2	Annual building energy consumption normalized by floor area, derived from Energy Use Intensity results	Minimize
	Hours Outside Comfort (h/year)	O3	Total annual hours during which zone operative temperature exceeds the defined comfort range, derived from zone temperature curves	Minimize
	Peak Cooling Demand (kW)	O4	Maximum instantaneous cooling load required to maintain indoor thermal conditions, extracted from load duration curves	Minimize

. These objectives include minimizing annual solar heat gain (O1), annual energy use intensity (O2), annual hours outside the thermal comfort range derived from zone operative temperature curves (O3), and peak cooling demand extracted from load duration analysis (O4). Collectively, these metrics capture complementary dimensions of façade thermal performance, encompassing long-term energy efficiency, solar-driven thermal loads, temporal thermal stability, and sensitivity to extreme cooling conditions. Façade design was parameterized using two primary decision variables: tessellation pattern type and perforation ratio, enabling systematic comparison across geometric configurations.

To ensure robustness and transparency in the identification of high-performance façade solutions, a deterministic multi-objective optimization approach based on Compromise Programming was adopted. This method prioritizes balanced performance by minimizing the overall deviation of each design alternative from the ideal solution across all objectives, thereby avoiding bias toward any single performance criterion [51]. By explicitly accounting for trade-offs among annual solar heat gain, energy use intensity, hours outside thermal comfort, and peak cooling demand, Compromise Programming provides a clear and interpretable ranking of discrete façade configurations. This approach is particularly suited to post-simulation decision-support analyses involving a finite set of design alternatives, as in the present study. The explicit definition of decision variables and performance objectives in Table 3 ensures clarity, reproducibility, and consistency within the optimization framework.

The optimization in this study is conducted as a post-simulation, data-driven process applied to a discrete set of design alternatives, rather than through continuous or iterative optimization. This approach enables systematic evaluation and ranking of façade configurations based on multiple performance criteria.

Compromise Programming Method

Compromise Programming is a deterministic multi-objective optimization method that identifies solutions achieving the best possible balance among conflicting objectives. The central idea is to define an ideal (utopia) point, representing the best attainable value of each objective when optimized individually, and then select the design alternative that minimizes its distance from this ideal solution. This approach is particularly suitable when no single objective dominates and when extreme underperformance in any one criterion should be avoided, as is often the case in building energy and thermal performance assessments.

The formulation of Compromise Programming is expressed as:

$$J_p\left(x\right)={\left(\sum _{i=1}^n{w}_i [z_{i }\left(x\right){]}^p\right)}^{{\displaystyle \frac{1}{p}}}$$

where $x$ represents a façade design alternative (defined by tessellation type and perforation ratio), $n$ is the number of objectives, $z_i\left(x\right)$ is the normalized deviation of objective $i$ from its ideal value, $w_i$ is the relative importance (weight) assigned to objective $i$, and $p$ defines the distance metric. In this study, the Euclidean distance ($p=2$) was adopted to penalize large deviations in any single objective and to favor solutions with stable, balanced performance across all criteria. Compromise Programming has been widely applied in energy and environmental decision-making problems, including building energy optimization, due to its robustness and transparency in handling trade-offs [52,53]. Therefore, in this study, the term “optimal” refers to the solution that minimizes the overall deviation from the ideal point across multiple objectives, rather than indicating absolute superiority in any single performance metric.

3. Results

3.1. Thermal Performance of the Base-Case Façade (No Shading Screen)

This subsection presents the thermal performance results of the base-case façade configuration without any external shading, which serves as the benchmark scenario for the comparative and optimization analyses. The results are derived from annual Climate Studio simulations using typical meteorological year data for Cagliari and are reported through four primary graphical outputs: Energy Flow, Energy Use Intensity, Load Duration Curves, and Zone Temperature Curves (Figure 4).

3.1.1. Energy Flow Analysis

The Energy Flow graph (Figure 4a) illustrates the monthly balance between heat gains and losses within the analyzed zone, disaggregated by internal gains (occupants, lighting, equipment), solar gains through glazing and façade elements, envelope heat transfer, ventilation, and mechanical heating and cooling. For the base-case façade, solar heat gain through glazing and façade surfaces is the dominant contributor to internal heat gains, particularly during the summer months. From May to September, solar gains exceed all other heat gain components, reaching peak values in July and August. These excessive gains directly drive the high cooling demand observed in the same period. In contrast, winter months are characterized by moderate envelope and ventilation losses, resulting in comparatively limited heating demand.

3.1.2. Energy Use Intensity

The Energy Use Intensity graph (Figure 4b) presents the monthly distribution of energy consumption per unit floor area, separated into heating, cooling, lighting, and equipment loads. The base-case façade exhibits a strongly cooling-dominated energy profile, consistent with Mediterranean climatic conditions. Cooling energy demand increases sharply from May and remains dominant until October, with peak values during July and August. Heating demand is confined mainly to winter months and remains comparatively low. Lighting and equipment loads remain relatively constant throughout the year and represent a smaller share of total energy use. These results indicate that cooling energy accounts for the majority of annual energy consumption, reflecting the combined effects of excessive solar heat gain and prolonged periods of elevated indoor temperatures.

3.1.3. Load Duration Curves (Heating and Cooling Demand)

The Load Duration curves (Figure 4c) show the cumulative number of hours during which heating and cooling loads exceed specific power thresholds over the year. This representation provides insight into both the magnitude and temporal persistence of thermal loads. For the base-case façade, the cooling load curve extends over a long duration, indicating that cooling is required for a substantial portion of the year. Peak cooling demand reaches approximately 2.6 kW, and non-zero cooling loads persist for several thousand hours annually. In contrast, heating loads are lower in magnitude and limited to a shorter duration, with a peak of approximately 1.9 kW. These results confirm that the building experiences frequent and sustained cooling demand, indicative of chronic overheating rather than isolated peak events.

3.1.4. Zone Temperature Curves (Thermal Comfort Conditions)

The Zone Temperature Curves (Figure 4d) illustrate the cumulative distribution of hourly operative temperatures throughout the year. This graph provides direct insight into indoor thermal comfort conditions and the frequency of temperature exceedances beyond acceptable comfort limits. The base-case façade experiences approximately 2006 h per year outside the comfort range, with the majority of these hours occurring above the upper comfort threshold of 26 °C. During summer months, operative temperatures frequently reach 26.5–27 °C, with maximum values approaching 28 °C. The temperature distribution indicates persistent overheating during occupied periods, highlighting the inability of the unshaded façade to maintain stable and comfortable indoor conditions under high solar exposure. In the baseline façade without shading, a total of 2006 annual hours fall outside the comfort range, of which approximately 1600 h are associated with overheating. This confirms that thermal discomfort is predominantly driven by summer conditions and excessive solar heat gains, establishing a critical baseline against which the performance of tessellated shading façades can be quantitatively evaluated.

3.1.5. Extraction and Synthesis of Thermal Performance Metrics

To quantify the thermal performance of the baseline and the proposed shading geometries, solar heat gain data was extracted from the simulation output variable Zone Windows Total Transmitted Solar Radiation Energy. This metric represents the cumulative solar energy transmitted through the glazing into the thermal zone, which was aggregated on both a monthly and annual basis to evaluate seasonal and total thermal impacts. Because the simulation engine provides energy results in Joules, the data was converted to kilowatt-hours (kWh)—the standard unit for building energy reporting—to facilitate a direct comparison with the Energy Use Intensity metrics. This conversion was performed by dividing the raw Joule values by 3,600,000, a factor derived from the 3600 s in one hour and the 1000 watts in a kilowatt (1 kWh = 3,600,000 J). By standardizing the transmitted solar radiation into kWh, the annual thermal reduction provided by the Islamic, Voronoi, and Folding patterns could be precisely measured against the unshaded baseline.

The total Energy Use Intensity was extracted from Energy Intensity graph to provide a comprehensive assessment of the building’s total energy performance across different facade configurations. Data for the individual energy components, including cooling, heating, lighting, and equipment, was extracted from the simulation results for each shading pattern (Islamic, Voronoi, and Folding) and perforation ratio (20%, 40%, and 60%). The Energy Use Intensity is expressed as the total energy consumed per unit area of the building over a one-year period, measured in kWh/m²·year. To arrive at the total energy use, the annual consumption values for all energy end-uses were summed and then normalized by the total floor area of the thermal zone. This normalization allows for an objective comparison between the baseline case and the protected scenarios, highlighting how specific geometric interventions optimize the balance between reduced solar heat gain and the resulting demands for artificial lighting and mechanical cooling.

Thermal comfort conditions were assessed using operative temperature-based indicators. Hourly operative temperature exceedances were extracted directly from the Zone Temperature Curves, quantifying the number of occupied hours outside the defined comfort range. The Zone Temperature Curves were further used to examine the annual distribution of indoor operative temperatures, enabling a temporal assessment of overheating and underheating periods without reliance on comfort indices requiring additional behavioral or physiological assumptions. Peak Cooling Demand refers to the maximum instantaneous cooling load required to maintain indoor thermal comfort during the hottest conditions of the year, as directly extracted from the Load Duration graph.

Table 4. Extracted thermal performance metrics for the base-case façade, including numerical values derived from Climate Studio simulation outputs and their corresponding interpretation.

Metric	Numerical Value	Source/Graph	Explanation/Interpretation
Annual Solar Heat Gain (kWh)	755.84	Zone Windows Total Transmitted Solar Radiation Energy	Excessive summer solar gains dominate the internal heat balance and drive cooling demand.
Total Energy Use Intensity (kWh/m2·year)	162	Energy Use Intensity	High annual energy demand, with cooling.
Hourly Operative Temperature h/year > 26 °C	Total ≈ 2006 Overheating ≈ 1600	Zone Temperature Curves	Hourly Operative Temperature
Peak Cooling Demand (kW)	2.6	Load Duration	Reflects the building’s sensitivity to solar heat gains, envelope performance, and internal loads

summarizes the thermal performance metrics extracted from the Climate Studio simulation outputs for the base-case façade. Each metric is reported together with its numerical value and the corresponding simulation graph used for extraction, enabling transparent benchmarking of the unshaded façade and facilitating direct comparison with the tessellated façade configurations evaluated later in the study.

3.2. Thermal Performance of Voronoi Tessellated Façade Patterns

Figure 5 presents the thermal performance of Voronoi-based tessellated façades at perforation ratios of 20%, 40%, and 60%, allowing for direct comparison under identical simulation conditions. The figure integrates Energy Flow, Energy Use Intensity, Load Duration Curves, and Zone Temperature Curves for each configuration, providing a comprehensive evaluation of how increasing façade openness influences solar heat gain, cooling demand, and indoor thermal conditions. This consolidated representation enables systematic assessment of performance variations associated with perforation density and Voronoi geometry.

Comparative Analysis of Voronoi Perforation Ratios

Across all Voronoi configurations, the tessellated shading screen substantially modifies the building’s thermal behavior compared to the base-case façade. Energy Flow graphs indicate that solar heat gains increase progressively as the perforation ratio rises from 20% to 60%, confirming the importance of shading density in limiting solar penetration. The 20% configuration provides the strongest reduction in summer solar gains, whereas the 60% configuration permits significantly higher solar transmission, approaching base-case conditions during peak summer periods.

This trend is reflected in the Energy Use Intensity results, where cooling demand is lowest at 20% perforation, increases at 40%, and rises further at 60%, while heating demand remains largely unchanged across all cases. Load Duration curves similarly show that lower perforation ratios reduce both peak cooling loads and cooling duration, whereas higher perforation ratios weaken shading effectiveness and increase cooling demand.

Thermal comfort analysis reveals consistent patterns. The 20% perforation ratio achieves the greatest reduction in overheating hours, while 40% and 60% configurations show progressively higher overheating frequencies. Although all Voronoi façades improve thermal conditions compared to the base case, increased perforation reduces their effectiveness. Specifically, the 20% configuration records 2017 annual discomfort hours, including approximately 1400 overheating hours, compared to 2008 and 1450 overheating hours for 40%, and 2004 and 1500 overheating hours for 60%. In contrast, the unshaded baseline records 2006 discomfort hours, including approximately 1600 overheating hours. These results indicate that while Voronoi shading reduces overheating across all perforation ratios, increasing façade openness progressively weakens its capacity to mitigate solar-driven thermal discomfort.

Table 5. Extracted thermal performance metrics for Voronoi tessellated façade configurations with 20%, 40%, and 60% perforation ratios, including numerical values derived from Climate Studio simulations and their corresponding interpretation.

Metric	Voronoi 20%		Voronoi 40%	Voronoi 60%	Interpretation
Annual Solar Heat Gain (kWh)	289.62		408.95	526.77	Solar heat gain increases with perforation, indicating reduced shading effectiveness.
Annual Energy Use Intensity (kWh/m2·year)	156		157	158	Cooling energy demand increases as perforation ratio increases.
Hourly Operative Temperature h/year > 26 °C	Total Overheating	≈2017 ≈1400	≈2008 ≈1450	≈2008 ≈1500	Higher perforation leads to more frequent overheating.
Peak Cooling Demand (kW)	1.8		2	2.3	Cooling demand raises with increasing perforation.

summarizes the extracted thermal performance metrics for the Voronoi tessellated façade at different perforation ratios. Each metric is reported together with its numerical value and corresponding simulation output, enabling a direct comparison of solar heat gain, energy demand, and operative temperature exceedances across façade configurations. This consolidated representation provides a transparent quantitative basis for comparing the thermal behavior of Voronoi patterns across varying levels of façade openness.

3.3. Thermal Performance of Islamic Geometric Tessellated Façade Patterns

Figure 6 illustrates the thermal performance of Islamic geometric tessellated façades across perforation ratios of 20%, 40%, and 60%. The combined presentation of Energy Flow, Energy Use Intensity, Load Duration Curves, and Zone Temperature Curves enables direct comparison of the impact of façade openness on solar heat gains, cooling energy demand, and thermal comfort under Mediterranean climatic conditions. This unified visualization supports the evaluation of performance differences and trends associated with varying perforation levels.

Comparative Thermal Performance of Islamic Pattern Perforation Ratios

Across all Islamic geometric configurations, the tessellated shading screen significantly influences the building’s thermal behavior relative to the base-case façade. Energy Flow results indicate a progressive increase in solar heat gains as perforation rises from 20% to 60%, particularly during summer months. The 20% configuration provides the strongest reduction in solar gains, while the 60% configuration allows substantially higher solar penetration, approaching base-case conditions. Energy Use Intensity results reflect this pattern, with cooling demand lowest at 20% perforation and increasing at 40% and 60%, whereas heating demand remains largely unaffected. Load Duration curves further confirm that peak cooling demand and cooling duration increase with higher perforation ratios, indicating reduced shading effectiveness at greater façade openness.

Thermal comfort analysis shows that overheating remains the primary source of discomfort across all configurations, with its magnitude increasing as perforation increases. The 20% perforation ratio records 2013 annual discomfort hours, including approximately 1500 overheating hours, representing a clear improvement over the unshaded baseline. At 40% perforation, overheating increases to approximately 1550 h within 2008 total discomfort hours, indicating a modest reduction in thermal performance compared to the 20% case. The 60% configuration records 2003 discomfort hours, including approximately 1580 overheating hours, approaching baseline overheating levels. In comparison, the unshaded façade records 2006 discomfort hours, including approximately 1600 overheating hours. These findings indicate that Islamic tessellated shading façades reduce overheating relative to the baseline; however, increasing perforation progressively weakens their solar control effectiveness and reduces their thermal performance benefits.

Table 6. Extracted thermal performance metrics for Islamic geometric tessellated façade configurations with 20%, 40%, and 60% perforation ratios.

Metric	Islamic 20%		Islamic 40%	Islamic 60%	Interpretation
Annual Solar Heat Gain (kWh)	290.88		409.95	533.94	Solar heat gain increases as perforation increases, indicating reduced shading effectiveness.
Annual Energy Use Intensity (kWh/m2·year)	155		156	157	Cooling energy demand rises progressively with higher perforation ratios.
Hourly Operative Temperature h/year > 26 °C	Total Overheating	≈2013 ≈1500	≈2008 ≈1550	≈2003 ≈1580	Overheating hours remain high, with limited variation across perforation levels.
Peak Cooling Demand (kW)	2		2.4	2.5	Cooling demand raises with increasing perforation.

summarizes the extracted thermal performance metrics for the Islamic geometric tessellated façade configurations with 20%, 40%, and 60% perforation ratios. Each metric is reported together with its numerical value and corresponding simulation output, enabling a direct comparison of solar heat gain, energy demand, and operative temperature exceedances across perforation levels. This consolidated presentation provides a consistent quantitative overview of the thermal performance of Islamic tessellated façades across different degrees of façade openness.

3.4. Thermal Performance of Folded Origami Tessellated Façade Patterns

Figure 7 presents the thermal performance of the folding (origami-based) tessellated façade across perforation ratios of 20%, 40%, and 60%, evaluated under consistent simulation conditions. The results include Energy Flow, Energy Use Intensity, Load Duration Curves, and Zone Temperature Curves, allowing direct comparison of energy demand, thermal loads, and indoor temperature behavior. This integrated representation facilitates assessment of the influence of perforation ratio on the thermal effectiveness of the folding façade geometry.

Simulation Results for Folding Patterns

Across all perforation ratios, the energy flow diagrams indicate a seasonal shift from heating-dominated conditions in winter months to cooling-dominated conditions during summer, with internal gains from occupants, lighting, and equipment remaining relatively stable throughout the year. Increasing perforation ratio leads to a progressive change in the balance between solar heat gains and cooling losses, particularly during peak summer months. The monthly Energy Use Intensity profiles show a reduction in cooling demand during mid-season months for higher perforation ratios, while winter heating demand remains comparable across the three configurations. Load duration curves illustrate a decrease in peak cooling loads as perforation increases, with the 60% configuration exhibiting the lowest maximum cooling demand.

The analysis of zone operative temperature curves for the folded shading façade indicates that all perforation ratios are characterized by a comparable magnitude of annual thermal discomfort. The 20% and 40% perforation configurations each exhibit 2015 h outside the comfort range, while the 60% perforation case shows a slightly lower value of 2011 h. Across all folding configurations, the majority of discomfort hours are associated with overheating, amounting to approximately 1500 h for the 20% perforation, 1515 h for the 40% perforation, and 1530 h for the 60% perforation, confirming that cooling demand remains the dominant source of thermal discomfort within the folded façade system. When compared to the unshaded baseline façade, which records 2006 annual hours outside the comfort range, of which approximately 1600 h are related to overheating, the folded shading systems exhibit a slight increase in total discomfort hours. However, despite this marginal increase, all folding configurations demonstrate a consistent reduction in overheating-related hours relative to the baseline, indicating a measurable mitigation of summer overheating conditions through the application of folded shading screens.

The extracted thermal performance metrics for the folding façade pattern at different perforation ratios are summarized in

Table 7. Extracted thermal performance metrics for Folded-Origami tessellated façade configurations with 20%, 40%, and 60% perforation ratios.

Metric	Folding 20%		Folding 40%	Folding 60%	Interpretation
Annual Solar Heat Gain (kWh)	219.31		306.98	427.63	Overheating hours remain high, with limited variation across perforation levels.
Annual Energy Use Intensity (kWh/m2·year)	155		157	157	Cooling energy demand rises progressively with higher perforation ratios.
Hourly Operative Temperature h/year > 26 °C	Total Overheating	≈2015 ≈1500	≈2015 ≈1515	≈2015 ≈1530	Occupant dissatisfaction increases with higher perforation ratios.
Peak Cooling Demand (kW)	1.80		2	2.40	Cooling demand raises with increasing perforation.

. Each metric is reported together with its numerical value and corresponding simulation output, enabling direct comparison of solar heat gain, energy demand, and operative temperature exceedances across folding configurations.

3.5. Thermal Performance Compression Between All Patterns and the Base Case

Figure 8a synthesizes the Energy Use Intensity performance of the base case and all geometric façade patterns across three perforation ratios (20%, 40%, and 60%), decomposed by end-use components (cooling, heating, lighting, and equipment). As shown in the stacked bar chart (a), the base case exhibits the highest total Energy Use Intensity, primarily driven by cooling demand, while all tessellated shading configurations achieve a measurable reduction in overall energy use. Across all patterns, equipment and lighting loads remain constant, reflecting fixed internal gains, whereas variations in total Energy Use Intensity are almost entirely attributable to changes in cooling and, to a lesser extent, heating demand. Among the tested configurations, lower perforation ratios consistently yield lower cooling energy, with the 20% perforation cases showing the most pronounced reductions relative to the baseline. Differences between geometric patterns are low but systematic, indicating that façade geometry also influences solar control efficiency beyond perforation ratio alone.

On the other hand, Figure 9b highlights the trend of performance decay with increasing perforation, expressed as a gradual rise in total Energy Use Intensity moving from 20% to 60% openness. All geometric patterns follow a similar upward trajectory, confirming that increased façade openness leads to higher solar admission and cooling demand. However, the rate of Energy Use Intensity increase varies by pattern, with the Voronoi configuration exhibiting a slightly flatter slope compared to the Islamic and Folding patterns, suggesting greater robustness to increased perforation. Importantly, all tessellated façades remain below the baseline Energy Use Intensity across the full perforation range, demonstrating their effectiveness as passive shading systems. This combined representation clarifies both the absolute energy contributions of different end uses and the relative sensitivity of each geometric pattern to perforation, providing a clear quantitative foundation for subsequent optimization and comparative evaluation.

Figure 9a synthesizes annual solar heat gain, annual energy savings, and peak summer (July) solar gains for three geometric shading strategies—Islamic, Voronoi, and Folding—evaluated at 20%, 40%, and 60% perforation ratios, benchmarked against the unshaded baseline. Results indicate a consistent increase in annual solar heat gain with higher perforation ratios across all patterns, reflecting the direct relationship between open-area fraction and solar transmission. At lower perforation levels (20%), all shading configurations significantly reduce annual solar heat gain compared to the baseline, with the Islamic pattern exhibiting the lowest annual gain, highlighting its superior shading effectiveness. However, as perforation increases, differences between patterns diminish, and at 60% perforation, solar gains approach those of the unshaded case. Figure 9b further illustrates this trend, showing a near-linear increase in annual solar heat gain with perforation ratio for all geometries, while consistently remaining below the baseline reference. Notably, the Folding pattern allows higher solar transmission at equivalent perforation ratios, whereas the Islamic and Voronoi patterns demonstrate more controlled solar admission, particularly at intermediate perforations (40%).

Figure 10 illustrates the variation in peak cooling demand across the unshaded baseline façade and the three tessellated shading strategies (Voronoi, Islamic, and Folding) at 20%, 40%, and 60% perforation ratios. The baseline façade exhibits the highest peak cooling demand (≈2.6 kW), reflecting direct solar exposure and the absence of any shading moderation. All tessellated façades reduce the peak cooling load to varying degrees, confirming the effectiveness of geometric shading in attenuating extreme summer cooling requirements.

Among the tested configurations, the 20% perforation ratio consistently delivers the lowest peak cooling demand across all patterns. The Voronoi 20% and Folding 20% façades perform best, each reducing the peak cooling demand to approximately 1.8 kW, representing the largest reduction relative to the baseline. As the perforation ratio increases, peak cooling demand rises for all patterns, with the 60% perforation cases approaching baseline values, particularly for the Islamic and Folding patterns. Overall, these results indicate that denser shading geometries with lower perforation ratios are most effective in limiting peak cooling demand, with the Voronoi and Folding patterns at 20% perforation emerging as the most balanced solution in terms of peak load reduction.

Figure 11a demonstrates the impact of tessellated shading patterns on annual thermal discomfort, with a specific focus on overheating, which constitutes the primary objective of this study. It shows that the total number of hours outside the comfort range remains relatively stable across all façade configurations when compared to the baseline, indicating that the application of shading screens does not substantially alter overall comfort exceedance. However, a clear redistribution between overheating and overcooling hours is observed. In all tessellated patterns and perforation ratios, overheating hours are consistently reduced relative to the unshaded baseline, confirming the effectiveness of façade shading in mitigating summer overheating conditions. The largest reduction in overheating hours is achieved by the Voronoi pattern at a 20% perforation ratio, followed closely by the folding pattern at the same perforation level, indicating that denser shading geometries are more effective in limiting excessive solar gains.

Figure 11b further highlights this behavior by isolating overheating-related hours and presenting their variation with increasing perforation ratio. The trend analysis shows a progressive increase in overheating hours as perforation increases across all geometric patterns, reflecting greater solar penetration at higher openness levels. Together, these results demonstrate that while overall comfort exceedance is largely unchanged, tessellated shading façades significantly reduce overheating-driven discomfort, with lower perforation ratios providing the most effective thermal control.

Although some absolute differences in performance metrics (e.g., 1–2 kWh/m²·year in Energy Use Intensity or approximately 100 h of thermal discomfort) may appear numerically small, it is important to interpret these values within the context of deterministic simulation outputs. The results obtained from Climate Studio (EnergyPlus engine) are not stochastic but represent fixed annual performance values under identical boundary conditions; therefore, conventional statistical measures such as standard deviation are not directly applicable. To ensure meaningful comparison, performance differences are evaluated using relative variations and normalized objective functions within the multi-objective optimization framework. For instance, a difference of 1 kWh/m²·year in Energy Use Intensity represents approximately 14% of the total variation range across all simulated configurations (155–162 kWh/m²·year), although it corresponds to less than 1% of the absolute annual energy demand. This indicates that even small numerical differences can be meaningful within the comparative evaluation framework. Similarly, a reduction of approximately 100 h of overheating represents a meaningful improvement in occupant thermal comfort over the annual occupancy period.

3.6. Multi-Objective Optimization

Following the evaluation of thermal performance across all tessellated façade configurations, a multi-objective optimization analysis was conducted to systematically identify high-performing design alternatives. Accordingly, four performance objectives were simultaneously considered, annual solar heat gain, annual energy use intensity, annual hours outside thermal comfort derived from zone operative temperature, and peak cooling demand, each formulated as a minimization goal. Compromise Programming was selected due to its ability to balance multiple competing thermal objectives without requiring subjective performance targets [54], making it particularly suitable for post-simulation evaluation of discrete façade design alternatives.

This optimization method was adopted as the multi-objective optimization method, as it enables the identification of façade configurations that achieve the best overall balance across competing objectives by minimizing the aggregated distance from an ideal performance point. This approach provides a transparent and objective ranking of design alternatives without requiring predefined performance targets or subjective weighting schemes, making it particularly suitable for comparative evaluation of discrete façade scenarios.

Table 8. Definition of design variables and performance objectives used in the multi-objective optimization analysis.

Design Parameters	O1 Solar Heat Gain	O2 Annual Energy Use Intensity	O3 Hours Outside Comfort (Overheating)	O4 Peak Cooling Demand
Voronoi 20%	289.62	156	1400	1.8
Voronoi 40%	408.95	157	1450	2
Voronoi 60%	526.77	158	1500	2.3
Islamic 20%	290.88	155	1500	2
Islamic 40%	409.95	156	1550	2.4
Islamic 60%	533.94	157	1580	2.5
Folding 20%	219.31	155	1500	1.8
Folding 40%	306.98	157	1515	2
Folding 60%	427.63	157	1530	2.4
Baseline	755.84	162	1600	2.6

summarizes the decision variables and performance objectives used in the optimization, defined over two discrete design variables, façade geometric typology and perforation ratio, representing the full set of tessellated shading configurations assessed in the thermal simulations.

Compromise Programming

Based on the application of the Compromise Programming approach, all façade configurations were evaluated simultaneously against the four defined performance objectives, annual solar heat gain, energy use intensity, hours outside thermal comfort, and peak cooling demand. The results identify the Voronoi tessellated façade with a 20% perforation ratio as the most balanced solution, exhibiting the smallest overall deviation from the ideal performance benchmark when all objectives are considered together. This configuration achieves a balanced reduction across long-term energy consumption, solar-driven thermal loads, temporal thermal discomfort, and peak cooling requirements, outperforming the remaining tessellated and baseline scenarios. The ranking further indicates that configurations with lower perforation ratios generally perform more favorably under this multi-criteria assessment, highlighting the importance of controlled façade openness in moderating both annual and peak thermal discomforts (

Table 9. Multi-objective optimization results using Compromise Programming (Lp = 2), reporting the raw objective values (O1–O4), their normalized counterparts, the aggregated compromise index, and the resulting ranking of façade design scenarios.

Design Scenario	O1	O2	O3	O4	O1 Norm	O2 Norm	O3 Norm	O4 Norm	Compromise Lp (p = 2)	Rank
Voronoi 20%	289.62	156	1400	1.8	0.131046	0.142857	0	0	0.096929	1
Folding 20%	219.31	155	1500	1.8	0	0	0.5	0	0.25000	2
Islamic 20%	290.88	155	1500	2	0	0	0.5	0.25	0.28736	3
Voronoi 40%	408.95	157	1450	2	0.353456	0.285714	0.25	0.25	0.287908	4
Folding 40%	306.98	157	1515	2	0.163402	0.285714	0.575	0.25	0.354068	5
Voronoi 60%	526.77	158	1500	2.3	0.573053	0.428571	0.5	0.625	0.536817	6
Folding 60%	427.63	157	1530	2.4	0.388273	0.285714	0.65	0.75	0.551677	7
Islamic 40%	409.95	156	1550	2.4	0.35532	0.142857	0.75	0.75	0.56384	8
Islamic 60%	533.94	157	1580	2.5	0.586416	0.285714	0.9	0.875	0.707309	9
Baseline	755.84	162	1600	2.6	1	1	1	1	1	10

).

Figure 12 visualizes the results of the Compromise Programming optimization across tessellation patterns and perforation ratios using a heat map representation of the aggregated compromise index (Lp = 2). Each cell reports the calculated compromise value, where lower values indicate closer proximity to the ideal solution across all four objectives (annual solar heat gain, energy use intensity, hours of overheating, and peak cooling demand). The results reveal a clear performance gradient: configurations with lower perforation ratios consistently achieve better overall performance, while increasing façade openness leads to progressively higher compromise values. Among all scenarios, the Voronoi pattern with a 20% perforation ratio exhibits the lowest compromise value, identifying it as the optimal façade configuration under this multi-criteria assessment. Folding patterns demonstrate moderate performance, particularly at 20% perforation, whereas Islamic geometric patterns show higher compromise values at increased perforation ratios, indicating reduced effectiveness in simultaneously minimizing all thermal objectives.

On the other hand, the radial chart (Figure 13) provides an alternative visualization of the Compromise Programming results, enabling a holistic comparison of façade patterns and perforation ratios across all thermal performance objectives. In this representation, the radial magnitude corresponds to the aggregated compromise index, where larger areas indicate poorer overall thermal performance and lower optimization ranking. The chart highlights Voronoi 20% as the most balanced configuration, while the unshaded baseline occupies the outermost position, confirming its inferior performance. Intermediate configurations reveal nuanced distinctions: Islamic 20% and Voronoi 40% rank closely in third and fourth positions with minimal performance difference, whereas Voronoi 60%, Folding 60%, and Islamic 40% form a cluster with comparable compromise values, indicating similar thermal behavior at higher perforation ratios. Overall, the radial comparison reinforces that, in addition to perforation ratio, where lower openness consistently yields better outcomes, the geometric typology itself plays a significant role, with Voronoi patterns achieving more balanced performance across multiple thermal criteria, while Islamic patterns generally exhibit weaker performance relative to the other tessellation strategies.

4. Discussion

4.1. Interpretation of the Findings

Although this study is based on simulation rather than experimental measurements, the adopted methodology is grounded in validated building energy modeling practices. Climate Studio utilizes the EnergyPlus simulation engine, which has been extensively validated and widely used in building performance research. The study employs a controlled parametric approach, where key design variables are systematically varied while maintaining consistent boundary conditions, ensuring the reliability and comparability of results. Furthermore, the integration of multi-objective optimization (Compromise Programming) extends the analysis beyond simple numerical simulation by enabling structured evaluation and ranking of design alternatives across multiple performance criteria. This approach provides a decision-support framework that is both reproducible and transferable to other contexts, thereby enhancing the scientific contribution of the work.

While the study does not employ iterative or evolutionary optimization algorithms, the adopted Compromise Programming method provides a structured multi-objective optimization framework that identifies balanced solutions among competing criteria. The optimization is conducted as a post-simulation, data-driven process applied to a discrete set of design alternatives, enabling systematic evaluation and ranking of façade configurations. This approach is particularly suitable for early-stage architectural design, where discrete design options are assessed and decision-making is guided by performance-based criteria rather than continuous parameter search.

The comparative analysis suggests that tessellated shading façades influence cooling-related thermal behavior relative to the unshaded baseline, with the extent of this influence varying according to both geometric typology and perforation ratio. Across the examined configurations, the presence of shading is generally associated with lower levels of annual solar heat gain and reduced peak cooling demand, indicating a consistent moderation of solar-driven thermal loads in the studied educational context. At the same time, the results point to a non-linear relationship between perforation ratio and performance, as increases in façade openness do not correspond to proportional improvements in thermal outcomes. Lower perforation ratios, particularly 20%, tend to provide more effective attenuation of solar gains while maintaining comparable indoor thermal conditions. Differences among tessellation typologies further suggest that thermal performance is shaped not only by the degree of openness but also by the geometric organization of façade patterns. Voronoi-based configurations show more stable performance across the evaluated metrics, while folding-based patterns exhibit comparable behavior at lower perforation levels but reduced effectiveness as openness increases. In contrast, Islamic geometric patterns appear more sensitive to higher perforation ratios, where increased solar transmission offsets potential shading benefits. Overall, these observations imply that the spatial distribution and morphological characteristics of openings play a substantive role in governing thermal response, reinforcing the relevance of geometry-informed façade strategies beyond perforation ratio alone.

An important observation from the results is the distinction between reducing summer overheating and affecting overall thermal comfort distribution. While all tessellated façade configurations demonstrate a clear reduction in overheating hours compared to the baseline, the total number of hours outside the comfort range does not decrease proportionally and, in some cases, slightly increases. This effect can be explained by the dual role of solar radiation: while shading reduces excessive heat gains during warm periods, it also limits beneficial solar gains during cooler or transitional periods, leading to more frequent cooler-than-comfort conditions. Therefore, reducing solar heat gain and overheating should not be interpreted as a direct improvement in overall thermal comfort, but rather as a trade-off between mitigating summer overheating conditions and maintaining balanced indoor conditions throughout the year. However, given the cooling-dominated Mediterranean climate of the case study, where overheating and cooling demand are the primary concerns, this trade-off is considered acceptable, as the reduction in cooling demand and peak thermal discomfort remains the dominant design objective.

Voronoi-based façades outperform Islamic geometric patterns primarily because of their non-uniform, adaptive distribution of openings, which leads to more effective solar modulation across changing sun angles and seasonal conditions. The irregular cell geometry of Voronoi patterns generates a heterogeneous shading field, creating a combination of self-shading, variable aperture depths, and staggered obstruction paths that collectively reduce direct solar penetration while maintaining diffuse daylight transmission. In contrast, Islamic geometric patterns are characterized by high symmetry and periodic repetition, which results in consistent alignment of openings and recurring direct solar paths; as perforation ratio increases, this regularity amplifies solar heat gain rather than attenuating it. Moreover, Voronoi geometries disrupt long-range visual and radiative continuity across the façade, limiting cumulative solar gains and reducing peak cooling demand more effectively than isotropic patterns. These results suggest that geometric anisotropy and spatial irregularity, rather than ornamental complexity or cultural geometry alone are critical drivers of thermal performance in perforated shading systems, particularly in cooling-dominated Mediterranean climates. While the analysis is based on a single façade orientation and case study, the results highlight fundamental relationships between geometry, perforation ratio, and solar control, which can inform façade design strategies in similar Mediterranean contexts.

To provide a deeper understanding of the observed performance trends, the relationship between façade geometry, perforation ratio, and thermal behavior can be interpreted through a cause–effect mechanism. As illustrated in Figure 14, tessellated façade geometry and perforation ratio directly influence shading density and solar obstruction characteristics. Lower perforation ratios increase surface coverage, thereby reducing incident solar radiation and limiting solar heat gain. This reduction in solar gains leads to lower indoor operative temperatures and decreased cooling demand. Conversely, higher perforation ratios increase façade openness, allowing greater solar penetration, which results in elevated indoor temperatures, increased overheating hours, and higher cooling energy demand.

In addition to perforation ratio, geometric variation (Voronoi, Islamic, and folding patterns) also affects the distribution and pattern of shading, thereby influencing thermal performance. As indicated in the results, the Voronoi pattern 40% demonstrates performance very close to the Islamic 20% configuration, ranking fourth with only marginal differences. This suggests that, when higher façade porosity is required, such as for improving daylight availability or daylight autonomy, irregular geometries like Voronoi can provide a more efficient balance between solar control and openness compared to more regular patterns such as Islamic pattern. These findings demonstrate that while perforation ratio remains the dominant parameter controlling thermal performance, geometric configuration acts as a differentiating and important variable in optimizing the trade-off between solar shading and environmental quality particularly at higher perforations.

The results indicate that perforation ratio (façade openness) is the dominant factor influencing thermal performance, as it directly controls the overall magnitude of solar radiation transmitted through the façade and, consequently, the cooling demand. At the same time, geometric variation also plays a meaningful role by shaping the spatial distribution and modulation of solar exposure. Its influence becomes more pronounced at higher perforation ratios, where increased openness reduces overall shading density and amplifies the effect of geometric morphology. Therefore, while perforation governs the primary thermal response, geometry acts as a differentiating parameter, particularly under high-porosity conditions.

4.2. Contribution of the Study in Respect to Prior Literature

The findings of this study both align with and extend existing research on performance-driven façade design by addressing several limitations identified across the literature. Earlier works have consistently established the façade as a critical mediator of environmental exchange and energy performance, with particular emphasis on passive shading strategies to reduce cooling demand in warm climates. Studies such as [55] demonstrate that lattice and double-skin façades can significantly influence thermal behavior, but also caution that geometric interventions may exacerbate overheating when not holistically integrated with ventilation and climatic context. The present study complements this insight by isolating the geometric and perforation effects of tessellated screens under controlled conditions, showing that geometry and openness must be jointly optimized to mitigate cooling-driven thermal discomfort rather than assumed to be inherently beneficial.

Much of the recent literature on parametric façades has prioritized daylighting and visual comfort as primary performance objectives. Rule-based and parametric approaches including Mashrabiya-inspired systems [56,57] and culturally derived folding patterns [27,58] demonstrate the architectural and visual potential of geometric articulation. However, thermal performance in these studies is often secondary, inferred qualitatively, or evaluated through limited indicators. In contrast, the present work foregrounds cooling-dominated thermal metrics, solar heat gain, energy use intensity, overheating hours, and peak cooling demand—thereby repositioning tessellation from a primarily visual or symbolic device to a quantitatively verifiable thermal strategy, particularly relevant for educational buildings with daytime occupancy peaks. Furthermore, consistent with prior studies highlighting the role of advanced and vernacular façade strategies in optimizing energy performance and visual permeability [59], this study confirms the effectiveness of façade design in reducing solar heat gain and cooling demand. However, it further contributes by providing a systematic, comparative evaluation framework that quantifies the influence of tessellated geometries and perforation ratios across multiple thermal performance indicators.

Comparative thermal assessment across fundamentally different tessellation logics remains rare in prior research. While biomimetic and irregular geometries have been shown to influence solar penetration and heat transfer behavior [60,61], these studies typically focus on component-level physics or solar-ray incidence rather than building-scale energy outcomes. The present study bridges this gap by demonstrating that geometric irregularity, exemplified by Voronoi tessellations, translates into consistently superior thermal performance at the building level, outperforming both culturally regular (Islamic) and structurally folded patterns under identical climatic and operational conditions. This finding provides empirical support for the hypothesis that anisotropic and non-periodic geometries more effectively disrupt cumulative solar heat gain and peak cooling loads than symmetric, repetitive patterns.

Research on origami and folding façades has highlighted their adaptability, self-shading capacity, and potential for kinetic operation or energy harvesting [62,63,64]. However, these studies often emphasize dynamic mechanisms, deployability, or photovoltaic integration rather than static thermal optimization. By evaluating folding patterns as fixed tessellated shading screens and comparing them directly with other geometric typologies, this study reveals that folding-based façades offer moderate thermal benefits at low perforation ratios but lose effectiveness as openness increases, a nuance that is largely absent from existing folding-façade literature.

Multi-objective optimization frameworks are widely employed in façade research to navigate trade-offs between energy, comfort, and daylight, commonly through evolutionary algorithms such as NSGA-II and SPEA-2 [65,66,67]. While these approaches are powerful, they often rely on continuous parameter spaces, surrogate modeling, or computationally intensive workflows that are difficult to translate into discrete architectural decision-making. In response to concerns raised by [68,69] regarding complexity and early-stage applicability, the present study adopts a post-simulation Compromise Programming approach tailored to discrete façade typologies and perforation levels. This decision-oriented optimization framework enables transparent ranking without reliance on subjective comfort indices or iterative re-simulation, making the results directly interpretable for design practice.

Finally, recent efforts emphasizing real-world validation and dynamic behavior, such as in situ glazing evaluation [70] and real-time bioreactor façades [71], underscore the variability of façade performance under changing conditions. While these studies advance monitoring techniques, they focus primarily on material systems rather than geometric shading strategies. By contrast, the present research demonstrates that even static tessellated screens, when systematically evaluated and optimized, can produce meaningful reductions in cooling-related thermal discomfort and peak demand, reinforcing the relevance of geometry-driven solutions within broader sustainability and resilience agendas.

Overall, this study contributes a missing link in the literature by integrating comparative tessellation typologies, cooling-focused thermal metrics, and a transparent multi-objective decision framework within an educational building context. In doing so, it advances façade research beyond isolated geometric explorations or single-metric optimization, offering a transferable and practice-oriented methodology for climate-responsive façade design. Beyond the specific configurations tested, the study contributes a generalizable methodological framework that integrates parametric design, simulation-based evaluation, and multi-objective optimization for façade performance assessment. This framework can be extended to additional variables, climates, and building types in future research.

4.3. Implications for Architectural Design and Energy Policy

The results of this study have direct implications for architectural design practice and environmental policy, particularly in the context of educational buildings in cooling-dominated climates. The consistent superiority of low-perforation tessellated façades, especially Voronoi-based configurations, demonstrates that façade shading should be conceived not merely as a decorative or secondary layer, but as a primary climate-responsive design strategy capable of substantially reducing solar heat gain and peak cooling demand. For architects and façade engineers, the findings suggest that prioritizing controlled façade openness and spatially irregular shading geometries can yield measurable energy and comfort benefits without increasing total hours of thermal discomfort. From a policy perspective, these results support the integration of geometry-based solar control criteria into building energy regulations and design guidelines, complementing existing prescriptive measures such as window-to-wall ratios and shading coefficients. In educational facilities, where occupancy schedules and cooling loads are highly synchronized, the reduction in peak cooling demand identified in this study is particularly relevant for HVAC sizing, operational cost reduction, and grid resilience. Collectively, the study provides evidence-based guidance that can inform façade design standards, retrofit strategies, and performance-driven design review processes aimed at improving thermal efficiency and sustainability in public educational buildings.

4.4. Limitations and Future Research Directions

While this study provides a comprehensive assessment of tessellated façade performance, several limitations should be acknowledged, primarily arising from the defined scope of the research and offering clear opportunities for future investigation. Although experimental validation was beyond the scope of this study, the use of a validated simulation engine ensures the robustness and reliability of the results.

First, the analysis focuses on a single educational building and a Mediterranean climate context, which, while appropriate for cooling-dominated conditions, limits the direct transferability of results to other climatic regions. Future studies could extend the framework to multiple climates and building typologies to evaluate climate-specific façade responses. In addition, the analysis is limited to a single façade orientation (east-facing), and future studies could investigate other orientations to assess orientation-specific performance variations.

Second, the investigation considers a discrete set of perforation ratios and tessellation typologies, selected to ensure methodological clarity and computational feasibility. Subsequent research could explore a broader and continuous design space, incorporating additional geometric parameters such as panel depth variation, orientation sensitivity, and adaptive or responsive façade behaviors. Future work could also extend this approach by exploring continuous parameter variation or higher-resolution sampling of perforation ratios using automated optimization or surrogate modeling techniques.

Third, thermal performance was evaluated using energy- and temperature-based metrics without explicitly modeling occupant-driven comfort indices, which were intentionally excluded to avoid reliance on uncertain behavioral assumptions. Future work could integrate dynamic occupancy schedules, adaptive comfort models, or monitored post-occupancy data to refine comfort assessments.

Fourth, the study does not include an economic evaluation of the proposed tessellated façade systems. While the results demonstrate clear differences in thermal performance, the associated costs of fabrication, installation, and maintenance were not assessed. In practice, these costs are expected to vary across different tessellation types, particularly due to differences in geometric complexity and constructability. For instance, more complex geometries such as Voronoi and folding-based patterns may require advanced digital fabrication techniques, potentially increasing production costs, whereas more regular and modular configurations, such as Islamic geometric patterns, may offer greater cost efficiency. Future research should integrate economic assessment methods, such as life-cycle cost analysis or cost–performance optimization, to support more comprehensive decision-making in façade design.

Finally, the optimization was conducted as a post-simulation decision-support process based on precomputed scenarios, rather than through real-time iterative simulation. Future research could couple advanced surrogate modeling or real-time co-simulation platforms to enable continuous optimization workflows.

It should be noted that the assessment of operative temperature exceedance in this study reflects conditions of thermal discomfort rather than physiological heat stress. According to standards such as EN 16798-1 [50] and ISO 17772-1 [72], these thresholds correspond to reduced comfort categories rather than critical heat stress conditions, which would require evaluation using indices such as PMV (Predicted Mean Vote) [73,74]. Future research could extend this framework to investigate the relationship between building overheating and potential heat stress under varying occupancy and environmental conditions.

Collectively, these limitations do not detract from the validity of the findings but instead delineate a structured pathway for extending the proposed methodology toward broader applicability, higher resolution, and greater integration with real-world design, economic feasibility, and operational contexts.

5. Conclusions

This study demonstrates that tessellated façade shading systems can contribute to reducing solar heat gain and peak cooling demand in educational buildings in cooling-dominated climates, with the magnitude of reduction varying significantly across different geometric typologies and perforation ratios. Among the tested geometries, the Voronoi 20% configuration achieves the most balanced overall performance across multiple thermal criteria; however, it does not reduce total annual discomfort hours compared to the baseline. This indicates that its performance advantage lies in reducing solar heat gain and cooling demand rather than in improving overall annual thermal comfort. It is followed by folding-based patterns, whereas Islamic geometric patterns show comparatively weaker performance, particularly at higher perforation ratios. Across all typologies, lower perforation ratios (20%) prove most effective in mitigating overheating and reducing cooling demand, underscoring the importance of controlled façade openness in thermally stressed environments.

The research effectively addressed the core research questions by quantifying the thermal impact of tessellated façade patterns, identifying the best-performing typology, and determining the optimal perforation ratio. First, the comparative simulations confirmed that tessellated shading screens enhance thermal performance relative to an unshaded façade by reducing solar heat gain, cooling energy use, overheating hours, and peak cooling demand. Second, cross-pattern evaluation demonstrated that geometric logic, beyond perforation alone, plays a decisive role, with irregular Voronoi geometries outperforming regular and folded patterns under identical conditions. Third, the multi-objective decision-support optimization results indicate that low-perforation configurations achieve the most favorable balance among competing thermal performance criteria, with the Voronoi 20% case emerging as the top-ranked solution. Together, these findings provide clear, evidence-based answers to the research questions posed in this study.

Beyond the specific performance outcomes, the primary contribution of this study lies in its integrative methodological approach and its reframing of tessellated façades as performance-driven environmental systems rather than purely aesthetic or cultural devices. By combining parametric façade modeling, multi-metric thermal evaluation, and a transparent post-simulation multi-objective decision-support framework, the study provides a replicable and systematic method for evaluating and ranking discrete façade alternatives in early-stage design. Rather than identifying a universally optimal solution, the approach enables informed decision-making by revealing trade-offs among competing performance criteria. In this context, the findings demonstrate that while perforation ratio governs the primary thermal response, geometric variation can be strategically leveraged as a differentiating parameter—particularly under higher porosity conditions—thereby bridging the gap between computational analysis and practical façade design decision-making.

Although the study is grounded in a single educational building located in a Mediterranean climate, the insights gained are transferable to other cooling-dominated and mixed climates where solar control and peak cooling reduction are critical design concerns. The underlying physical principles governing solar interception, shading continuity, and radiative disruption are climate-agnostic, even though their quantitative expression varies by location. Moreover, the methodological framework can be readily extended to other building types, façade systems, and climatic contexts. Ultimately, this research contributes to the broader pursuit of climate-responsive architecture by demonstrating how geometry-informed façade design can support energy efficiency, thermal resilience, and sustainable educational environments in an era of increasing climatic stress.